Random and non-random (deterministic) dynamical systems governed by systems of delay ordinary differential equation (ODE) and delay stochastic differential equation (SDE) play an important role in the investigation of infectious disease systems. For instance, several ecological models consist of delay SDE and delay ODE representing the dynamics of infectious diseases of humans, animals, and plants (cf. [11,12,18,27,29,30,32,41,42,47]); they also represent dynamics of malicious signals (or worms) on wireless sensory networks (WSN)(cf. [20,56]); they represent dynamics of rumors and malicious ideas on complex social networks (cf. [19]), etc. Note that the term disease is used both technically and metaphorically to represent a condition that is prevalent in a system.